Take the 2-minute tour ×
Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. It's 100% free, no registration required.

I need to calculate the skewness and kurtosis of 2 asset portfolio, can someone please help me with the formulas and definition of terms? Thank you.

I have been using the matrices method and I am not sure if that is correct.

share|improve this question
    
Can you give us some more information such as what you are trying, what data you have and what you hope to achieve? –  Patrick Burns Oct 29 '11 at 10:26
    
Please define the "matrices method". –  Shane Oct 29 '11 at 19:10
    
What's wrong with the standard definitions available on wikipedia? –  Tal Fishman Oct 29 '11 at 23:02
    
I am trying to look at the effects on the VIX index on hedge funds and I need to calculate the resulting skewness and Kurtosis when different weights of the VIX is added to the hedge fund portfolio. So the HF returns is considered as stock A and the Vix is considered as stock B. I have been using the matrices method to calculate the comoments. I need to find a formula to calculate the portfolio skewness and kurtosis.I have already calculated the skewness and kurtosis of each variable on their own. TY –  user1642 Nov 8 '11 at 15:48
    
I am trying to look at the effects on the VIX index on hedge funds and I need to calculate the resulting skewness and Kurtosis when different weights of the VIX is added to the hedge fund portfolio. So the HF returns is considered as stock A and the Vix is considered as stock B. I have been using the matrices method to calculate the comoments. I need to find a formula to calculate the portfolio skewness and kurtosis.I have already calculated the skewness and kurtosis of each variable on their own. TY –  user1642 Nov 8 '11 at 15:49

2 Answers 2

"Skewness" quantifies how asymetric a distribution is about the mean. "Kurtosis" quantifies how peaked or flat the distribution is.

Skewness is defined as:

$E[ (X - mean)^3 ] = \frac{(\sum (x_i - x_{mean})^3 )}{N}$

and Kurtosis as:

$E[ (X - mean)^4 ] = \frac{(\sum (x_i - x_{mean})^4 )}{N}$

where X is your distro values (x_1, x_2, ... x_N), mean is the average of your distro values X (x_mean, a constant) and E[f(X)] is the Expectation of f(X) - i.e the mean of f(X).

So now you need to define your distributions. To be honest I don't know what the standards are for a given asset, but I imagine that if your asset price movements are ~ lognormal then you'll be wanting the daily (or whatever) percentage change in the value of the portfolio. These daily %age changes define your distribution X. Of course you'll need to consider how far back in time you go: 1 month data? 1 year?. So each daily %age change is your x_i. Calc the mean (probably close to zero), then your Skewness and Kurtosis per the formulas above.

share|improve this answer
1  
More specifically, skew (the third moment about the mean) is defined as $\int(x-\mu)^3 p(x)$ and kurtosis (the fourth moment about the mean) as $\int(x-\mu)^4 p(x)$. –  strimp099 Oct 29 '11 at 14:23

Assuming you have return time series $$ r_1(1), r_1(2), \ldots, r_1(T) \qquad \text{and} \qquad r_2(1), r_2(2), \ldots, r_2(T) $$ for the 2 assets and asset weights $w_1$ and $w_2$, we can follow the calculation of the $N$-asset portfolio skewness laid out in another answer for a similar question.

To extend it to include portfolio kurtosis, we need the co-kurtosis tensor $$ K_{ijkl} = E \left[ r_i \times r_j \times r_k \times r_l \right] = \frac{1}{T} \sum_{t=1}^T r_i(t) \times r_j(t) \times r_k(t) \times r_l(t) $$ and moment $$ m_4 = \sum_{i=1}^N \sum_{j=1}^N \sum_{k=1}^N \sum_{l=1}^N w_i w_j w_k w_l K_{ijkl} \quad, $$ then we can calculate portfolio kurtosis as $$ K_p = \frac{1}{\sigma_p^4} \left[ m_4 - 4 m_3 m_1 +6 m_2 m_1^2 - 3m_1^4 \right] \quad. $$

In the 2 asset portfolio case, the calculations of the higher-order tensors are not so daunting, as for the $2 \times 2 \times 2$ co-skewness tensor we only need to calculate $$\begin{split} S_{111} & \\ S_{112} &= S_{121} = S_{211} \\ S_{122} &= S_{212} = S_{221} \\ S_{222} & \end{split}$$ and for the $2 \times 2 \times 2 \times 2$ co-kurtosis tensor we only need to calculate $$\begin{split} K_{1111} & \\ K_{1112} &= K_{1121} = K_{1211} = K_{2111} \\ K_{1122} &= K_{1212} = K_{1221} = K_{2112} = K_{2211} \\ K_{1222} &= K_{2122} = K_{2221} = K_{2212} \\ K_{2222} & \end{split}$$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.